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CIRM Luminy « Quest-ce que la géométrie aux époques modernes et contemporaines ?» 16-20 avril 2007 « Die Wahrheit des Bildes ist unabhängig von dem Grade der Feinheit des Bildes » or should we construe Riemanns « Physical Space » as a Mechanical Ether ? Ivahn Smadja (Université de Caen) The close connection between natural philosophy and foundations of geometry in the early thinking of B. Riemann is a well known fact. But it is usually accounted for in terms of transition from a sketchy mechanical ether theory to mature geometrical thought, at the cost of downplaying or even putting aside some unexplained features of Riemanns approach. Bringing together the unpublished philosophical manuscripts Riemann wrote a few months before the inaugural dissertation on the foundations of geometry might provide a somewhat different view of the process by which geometrization of physical theory led to rethinking geometry.

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1. Naturphilosophie 1.1. Unification of Natural Phenomena 1.1.1. Physiology, Psychology and Physics 1.1.2. Gravity explained by Ether Pressure : Euler [1744] 1.1.3. Potential theory : from Mathematical Device to Physical Theory. 1.2. Open Questions 1.2.1. Mechanical Models : Continuous Ether or Punctiform Ether ? 1.2.2. What is Pressure ? Lagrangian Mechanics versus « Mécanique physique » 1.2.3. « die kühne Hypothese des Verschwindens der Materie » In the Neue Mathematische Principien der Naturphilosophie (march 1853), Riemann works out a mechanical hypothesis taken from Euler but recasts it by transposition of the concepts of Eulerian hydrodynamics to an upper level of physical reality. Euler explains gravity by the pressure of an infinitely fluid surrounding ether of constant density, but according to his original hypothesis, the fluid is compressible, and there is a longitudinal motion of ether in the direction of the pressure gradient (a longitudinal wave as for sound, Euler says). In the Riemanns hypothesis on the contrary, the fluid is supposed « incompressible and without inertia ». So there is a clear cut alternative : either ascribing the motion of ether to its elasticity and therefore regarding incompressibility as a limit case of (very little) compressibility, or taking into account different levels of characterization of physical reality.

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Gravity explained by Ether Pressure : L. Euler Anleitung zur Naturlehre Chapter 19. Of Gravity and of the Forces which exert themselves upon Celestial Bodies. §. 140. « Gravity comes from inequal pressure of the ether, which increases with the distance from Earth; therefore the bodies are, as from themselves, pushed toward the Earth, and the supplement of those pressure forces equals the weight of the body ». §. 141. « Gravity exert itself on bodies only as far as they are composed of gross matter, and the greater is the space filled with gross matter, the greater is the weight of the Body, that is the weight of the Body behaves as its true magnitude. » §. 142.« As experience teaches us that the weight of a body decreases with the square of the distance, as it recedes from the center of the Earth, it should be supposed, in order to explain this fact, that the pressure of the ether against the center of the Earth is such that the decrease in pressure is inversely proportional to the distance to this center ». b b a a Earth C C CP=x r Pressure on face aa Pressure on face bb Force exerted on aa Force exerted on bb P The magnitude h expresses the elastic force of the ether, where it is in equilibrium, and the magnitude the elastic force which the ether exerts at a distance CP=x of the center of the Earth.

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C.F. Gauss & W. Weber Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte (1839-40) Supposing et denote masses, quantities of magnetic or electric fluid, there is one and the same form of law for gravitational, magnetostatic or electrostatic phenomena, The key idea of potential theory is dissociate the two components of the force, that is the mass of the test body, and the accelerative force [beschleunigende Kraft] The components of the accelerative force of intensity toward are equal to and happen to be partial derivatives of a scalar magnitude, the potential, The distribution of mass might now be discrete or continuous, and accordingly the potential will be a sum or an integral Divergence Theorem (cf. Allgemeine Lehrsätze, § XXII.) in case S is a sphere and then the divergence theorem gives F F r x z y (a,b,c) F n Cn C S C= V

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Lagrangian Mechanics versus « Mécanique physique » What is pressure? Binding force or mean molecular repulsion? a. Lagrangian multipliers and fictitious forces Let P, Q, R … be forces acting upon a system of material points, and p, q, r, … the distances from the points upon which the forces are exerted to the respective centers of forces. The value of p, q, r, … may be regarded as functions of the variables. The total differential dp, dq, dr, … as functions of are then substituted in the general formula of equilibrium The linear equation in thus obtained might then be simplified using the following « equations of conditions » between the variables L=0, M=0, N=0, … and a system of independent differentials is obtained after elimination in the system of differential equations dL=0, dM=0, dN=0, …. But there is another way around, by adding from the very outset the « equations of condition » multiplied by indeterminate quantities to the general formula of equilibrium Therefore adding to actual forces purely fictitious ones, depending on the nature of binding conditions of the mechanical system and on the expression of the Lagrangian multipliers. Either, the only forces considered are the actual ones, but the restrictions that the binding conditions impose on virtual displacements are taken into account, or those restrictions are removed but some corresponding fictitious forces must come into the picture.

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2.Molecular Mechanics 2.1. Capillarity 2.2. « A new fundamental Principle of Mechanics » 2.2.1. Least constraints [Gauss 1829]. 2.2.2. A Gaussian Analogy 2.3. Riemanns memoir on Molecular Mechanics 2.4. Riemannian Epistemology 2.4.1. Concept-formation and Hypotheses 2.4.2. « Bildtheorie » 2.4.3. Antinomies In Molecularmechanik, Riemann presents a conception of mechanics along the lines of the principle of least constraint enunciated by Gauss [1829] who hit upon it in the context of his researches devoted to the theory of capillar action. This gaussian line of thought concerning physical theory allows Riemann to surmount the opposition between continuous medium theory and punctiform ether theory which was a locus classicus of mechanical model building in the 1840s and the beginning of the 1850s continental physics. Actually, Riemann hierarchizes the degrees of fine- and coarse-grainedness of the theoretical picture, so that the hypothesis of a continuous medium may not be incompatible with punctiform ether models favored as more explanatory in those years before British ideas penetrated German physics a few decades later. Support for such level-diffentiated pictures may be adduced from epistemological writings where Riemann hints at some sort of « Bildtheorie ».

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Theory of capillar Action I. Newton, P.S. Laplace, Th. Young, S.D. Poisson, C.F. Gauss Capillar phenomena, such as countergravity progression of liquids in narrow tubes, are essentially due to short range molecular forces. (e.g. according to Laplaces explanation, the curvature of the menicus implies a supplement of molecular attractionson the particle E, and therefore its rising up above the level of D. E A BC D Laplace N Poissons paradox. Points in the neighbourhood of N cannot be in equilibrium (1) (2) (3) (4) (5) (0)air water

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Gauss [1830] builds the whole theory of capillar action on the notion of surface tension and surface energy as in the theory of Th. Young, but gives its full general scope in formulating it in terms of the Calculus of variations. The limiting surface between different media has a potential energy of its own which depends only on the area of the surface, such that there is equilibrium when the sum is minimum, whereis the potential energy per unit of area, the area of contact between media i and j, and potential energy.

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The principle of least constraint. Gauss (1829) The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of N masses is the minimum of the quantity for all trajectories satisfying any imposed constraints, where and represent the mass, position and applied forces of the k th mass. Gauss' principle is equivalent to DAlembert principle.

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3.Gravitation and Light 3.1. Eulerian hydrodynamics and Physicalization of Vector Fields 3.2. Causality 3.3. Adding Vector Fields In Gravitation und Licht (december 1853), Riemann assumes that an overall explanation of gravitation and light phenomena should be sought in the « form of motion » of a kind of substance « continuously extended in the infinite space », which he calls Stoff, raumerfüllende Stoff or simply Raumstoff, rather than Aether. But at the same time, he makes a sharp distinction between two superposed though distinct levels of causality. On the one hand, the laws that govern the motions of this substance, the Stoffbewegungen, that one ought to assume in order to explain the phenomena, and on the other hand, the ultimate causes, the Ursache, by which the very motions of this substance could be explained. According to Riemann, the true causes of the propagation of gravitation and light are largely unknown to us, and the question of the actual mechanical constitution of the ether, wether continuous or molecular, is a metaphysical concern, but this theoretical limitation of physical pictures doesnt prevent us to build a coherent mathematical treatment of the unification of natural phénomena by adequately superposing vectors fields. Construing potential theory in the light of an epistemological theory of causality, Riemann thus endows his Raumstoff with an equivalent of the quasi- corporeity Gauss endowed surfaces when tracing the way to intrinsic geometry.

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4. Raumstoff and Intrinsic Geometry 4.1. Quadratic Forms and Main Dilatations 4.2. « Surfaces are Bodies from which one dimension disappears » 4.3. Riemanns Hypothesis [1853] 4.4. Curvature of Space : « After the manner of a wave » [Clifford 1876] A. Mechanical Ether Supposing two neighbouring points P(x,y,z) and Q(x+dx,y+dy,z+dz), the extension of a Stofftheilchen might be thought of as element of line ds separating the two points. If the propagation of gravitation and light is imputable to the elasticity of a mechanical ether, then the dilatation or contraction of the Stofftheilchen correspond to the passage from ds 2 = dx 2 + dy 2 + dz 2 to ds 2 = dx 2 + dy 2 + dz 2 that is from a quadratic form at time t to a quadratic form at time t+dt. According to this mechanical scheme, forces would propagate thanks to the fundamental elasticty of the medium. Q(x+dx,y+dy,z+dz) au temps t ds P(x,y,z) au temps t+dt y P(x,y,z) au temps t ds z x P(x+dx,y+dy,z+dz) au temps t+dt

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